Finding Domain and Range with Visual Representation in A Graphing Calculator

The concept of domain and range is incredibly fundamental, making it crucial to understand fully. It also has applications in the basics of Calculus concepts such as limits and continuity and simple differentiation.

Bonus fun fact: the SAT Math section often tests domain and range. Let’s get started with a few definitions! 🤓

What is Domain? 🌎

In simple terms, the domain of a function is a list of every possible input value, or x-value, that allows the function to be defined. To understand this more fully, let’s look at a few basic examples:

Example 1: Function Map

Image Courtesy of Wikipedia

In the image above, the shown function is defined for the x-values 1, 2, and 3. Thus, we say that the domain of this function is the set of {1, 2, 3}.

Example 2: y=x

Take a moment to picture the graph of this function in your head. For what values of x is this function defined?

y = x; Graph Courtesy of Desmos.

As it turns out, y = x is an infinite line in the coordinate plane, meaning that the line is defined for all real values of x. Whenever a function is defined for all real number inputs, we say that the domain of the function is ℝ. Thus, the domain of y = x is ℝ (another way of saying "for all real numbers").

Example 3: y = x^2

Again, try to first envision the graph of this parent quadratic function in your head. What are all the possible inputs of y = x^2?

Recall (and you'll also realize through your imagination) that any real number squared is defined. Again, the domain of y = x^2 is defined for ℝ.

y = x^2; Graph Courtesy of Desmos.

Example 4: y = 1/x

This example may seem slightly different and harder to visualize. If you have trouble picturing this in your head, it may be a good idea to pick a few values of x, find the y-value, and sketch it out! (That's right: channel your inner Picasso!) 🎨

y = 1/x; Graph Courtesy of Desmos.

Now, it's important to note that while the x-coordinates get increasingly closer to the y-axis on either side, we'll never be able to define the x-coordinates for a value of zero. Graphically, we can deduce that the domain of the graph of y = 1/x will include all real numbers EXCEPT FOR zero. In set notation, we write ℝ \ {0}.

Still stumped? Think of this answer from an algebraic standpoint: we know that a fraction is never defined if the denominator is zero. When we plug in zero to the equation y = 1/x, we get a zero in the denominator, so the function is not defined for an x-value of zero 😊

What is Range? 🎯

To recap, the domain is all the possible inputs of a function. The range, on the other hand, is all the possible outputs, aka y-values, of a function. Consider the following examples:

Example 1: Function Map

Image Courtesy of Wikipedia.

As established above, the domain of this function is {1, 2, 3}. Now, all we need to do is consider all the y-values that all possible x-values map to. We'll find that the range is then {a, b}.

Example 2: y = 1.2x+5

Again, first try to visualize the graph of this function. What are all possible values of y from this equation?

y = 1.2x + 5; Graph Courtesy of Desmos.

In this case, there is no value of y that is undefined, so the range of this function is all real numbers, or ℝ.

Example 3: y = x^2

To approach this, recall that it is impossible for the square of any real number to be negative. This means that the range of y = x^2 is the set of all nonnegative numbers. In interval notation, we write this as [0, ∞).

Try considering this graphically:

y = x^2; Graph Courtesy of Desmos.

On the graph of y = x^2, we can see that the y-values extend indefinitely (forever!) in the positive direction but have a minimum value of 0. In other words, y = x^2 never stretches below the x-axis, which reinforces our prior algebraic deduction of the range being [0, ∞).

Example 4: y = 1/x

Again consider the graph of this function:

y = 1/x; Graph Courtesy of Desmos

Notice that while the y-coordinates get increasingly closer to the x-axis from both top and bottom, the y-coordinates are never defined for a value of zero. Graphically, we can deduce that the range of the graph of y = 1/x is all real numbers excluding zero. In set notation, we write ℝ \ {0}. Feel like deja vu?

Again, consider this answer from an algebraic standpoint—it's never possible for a fraction to equal zero if the numerator is nonzero. This means that the function 1/x can never be equal to zero, no matter the value of x! 🤓

Takeaways & Strategies 🏹

There are several strategies that you can employ to find the domain and range of any function:

1. Graphical: First, try to visualize a graph of the function in your head and see if you can make any deductions about the possible x and y-values of the function. 💹

2. Algebraic: See if any properties of the specific function at hand allow you to rule out any x or y-value possibilities. 📝

Understanding the concept of domain and range is a MUST in advanced fields of mathematics that you’ll encounter later on. Kudos to you for mastering such an important skill as early as today! 👏